Can $h^{1, 0}$ and $h^{1, 1}$ jump for smooth projective surfaces over $\mathbb{Z}[1/N]$?

Let $$N$$ be a positive integer. Let $$f:X\to S=\mathrm{Spec}\:\mathbb{Z}[1/N]$$ be a smooth projective morphism of relative dimension 2 such that $$R^1f_*\mathcal{O}_X$$ and $$R^2f_*\mathcal{O}_X$$ are both locally free $$\mathcal{O}_S$$-modules. Can $$\mathrm{dim}_{k(s)}H^0\big(X_s, \Omega^1_{X_{s}/k(s)}\big)$$ and $$\mathrm{dim}_{k(s)}H^1\big(X_s, \Omega^1_{X_{s}/k(s)}\big)$$ depend on the closed point $$s\in S$$?

• The Euler characteristic of $\Omega^1_{X_s/k(s)}$ is constant (Hartshorne III 12). Jul 2, 2020 at 18:17
• You changed the question after I answered it. Please indicate the extent of such substantial edits in the future. Jul 2, 2020 at 19:22

Yes. An Enriques surface with classical reduction at $$p=2$$ gives such an example. See Illusie "Complexe de de Rham-Witt et cohomologie cristalline" Prop. II 7.3.8(b), p. 658.
• TBH, I don't know offhand if they exist over $\mathbf{Z}[1/N]$ specifically, not just some $\mathcal{O}_K[1/N]$ with $N$ odd and $K$ a number field unramified at $2$. Jul 2, 2020 at 20:29